Notice that both \(\frac{dx_i}{dt} \big \vert _{x_i=0}=0\)  and \(\frac{dy_i}{dt} \big \vert _{y_i=0}=0\)  for \(i=1,2\) , thus according to Theorem A.4 (p.423) in [1]}, we can conclude that the model (REF ) is positive invariant in \(\mathbb {R}^4_+\) . Now we can go ahead to show the boundedness of the system. First, we have the following inequalities due to the property of positive invariance:
\(\frac{dx_i}{dt}= r_i x_i\left(1 - \frac{x_i}{K_i}\right) - \frac{a_i x_i y_i}{1 + x_i}\le r_i x_i\left(1 - \frac{x_i}{K_i}\right)\) 
